Fall 2005
6.012 Microelectronic Devices and Circuits
Prof. J. A. del Alamo
Homework #2 ­ September 16, 2005
Due: September 23, 2005 at recitation ( 2 PM latest)
(late homework will not be accepted)
Please write your recitation session time on your problem set solution.
1. [20 points] Consider a p­type Si sample with a resistivity of 2 Ω.cm at 300 K. At 300K,
estimate the magnitude of:
1a. [5 points] Hole concentration, po .
1b. [5 points] Electron concentration, no .
1c. [5 points] What is an appropriate value for the hole mobility, µp ?
1d. [5 points] What is an appropriate value for the electron mobility, µn ?
State whatever assumptions you need to make.
2. [20 points] Consider a sample prepared from the same material as in problem 1. This
sample is at 300K. The geometry of the sample is such that it can be considered a one­
dimensional situation. At a certain location in this sample, we measure a drift current
density of 104 A/cm2.
J=104 A/cm2
2a. [5 points] Estimate the magnitude of the electric field at this location.
2b. [5 points] Estimate the relative contribution of electron and hole drift to the total
current.
2c. [5 points] Estimate the hole drift velocity.
2d. [5 points] Estimate the electron drift velocity.
3. [20 points] Consider a piece of n­type Si in thermal equilibrium at 300 K. In a region
defined by 0 ≤ x (µm) ≤ 10, there is a spatially varying electron concentration as sketched
below.
no (cm-3)
1017
no (x) = 1017−0.2x cm−3
1016
with x in µm
1015
1014
0
5
10
x (µm)
3a) [5 points] Derive an analytical equation for the minority carrier concentration in space.
Quantitatively sketch the result in a suitable diagram.
3b) [5 points] Derive an analytical equation for the electrostatic potential in space. Quan­
titatively sketch the result in a suitable diagram.
3c) [5 points] Derive an analytical equation for the electric field in space. Quantitatively
sketch the result in a suitable diagram.
3d) [5 points] Derive an analytical expression for the charge distribution that supports this
electric field. Quantitatively sketch the result in a suitable diagram.
4. [40 points] I­V characteristics of pn diode (cont.)
This exercise continues the pn diode characterization problem of homework #1. It utilizes
the data for the current­voltage characteristics for a pn diode that you obtained through the
MIT Microelectronics WebLab last week. In this exercise, your task is to derive a simple
equivalent circuit model for this pn diode. You should study Appendix A at the end of this
package.
Do the following:
4a) (10 points) Study the ideal model for the I­V characteristics of the pn diode in Ap­
pendix A. Devise a simple scheme to extract from the measured data the saturation
current, Is (in A) and the temperature of the diode, T (in K). Illustrate your proce­
dure graphically and give the extracted values. Give a higher weight to estimations of
Is that come from the ideal looking portion of the forward branch of the I­V character­
istics where the accuracy of the model is most important. Compare the temperature
that you extract with the temperature of the lab as measured by WebLab. Comment
on discrepancies.
4b) (10 points) Compare the experimental I­V characteristics with those predicted by the
ideal theoretical model by graphing them together. For this, you will need to use
MATLAB, a spreadsheet program, or some other mathematical program with graph­
ing capabilities.
Turn in the following graphs:
graph 1: Linear plot of I­V characteristics (V in x axis in linear scale, I in y axis
in linear scale). Show experimental data points with symbols and ideal model
with continuous line. Print out this graph.
graph 2: Semilogarithmic plot of I­V characteristics (V in x axis in linear scale, I in
y axis in logarithmic scale). Show experimental data points with symbols and
ideal model with continuous line. Print out this graph.
4c) (10 points) A more realistic model for a pn diode includes a parasitic series resistance, as
discussed in Appendix A. Using the values of Is and T derived in the previous section,
devise a simple scheme to extract from the measured data the series resistance, Rs (in
Ω), of the diode. Illustrate your procedure graphically and give the extracted value.
4d) (10 points) Compare the experimental characteristics with those predicted by the the­
oretical model by graphing them together. Plotting the I­V characteristics of the
model that includes series resistance is a bit tricky because I is on both sides of the
equation. A good way to do it is to solve for V , then compute V vs. I, and finally
plot I vs. V .
Turn in the following graphs:
graph 3: Linear plot of I­V characteristics (V in x axis in linear scale, I in y axis
in linear scale). Show experimental data points with symbols and second­order
model with continuous line. Print out this graph.
graph 4: Semilogarithmic plot of I­V characteristics (V in x axis in linear scale, I in
y axis in logarithmic scale). Show experimental data points with symbols and
second­order model with continuous line. Print out this graph.
Turn in these four graphs and comment on how the measured I­V characteristics of the
diode compare with those that you studied in 6.002.
The required graphs need not be too fancy, just simply correct and unambigous. They must
have proper tickmarks, axis labelling, and correct units.
Appendix A: DC I­V characteristics of pn diode
Ideal model
The ideal I­V characteristics of a pn diode are given by:
�
I = Is exp
�
qV
−1
kT
where Is is the saturation current, q is the electron charge (in this expression, q = 1 e ,where
e stands for electron, and NOT q = 1.6 × 10−19 C), T is the temperature (in K), and K is
Boltzmann constant (K = 8.62 × 10−5 eV /K).
Second­order model
”Real” diodes suffer from a number of parasitics. One of the most important ones is the
presence of parasitic series resistance, Rs. This reduces the voltage that is available to
the junction from an external one V to an internal one V − IRs . Hence, the DC I­V
characteristics of the diode are given by:
�
q(V − IRs )
I = Is exp
−1
kT
�
The I­V characteristics predicted by both models look as in the graphs below.
ideal
with series resistance
I
I
log |I|
IRs
+
1/Rs
Rs
V
ideal�
diode
-
0
0
linear scale
V
0
V
semilogarithmic scale
Figure 1: Sketch of I­V characteristics (ideal and with series resistance) of p­n junction in linear
and semilogarithmic scales.